Points on Computable Curves

The “analyst’s traveling salesman theorem” of geometric measure theory characterizes those subsets of Euclidean space that are contained in curves of finite length. This result, proven for the plane by Jones (1990) and extended to higher-dimensional Euclidean spaces by Okikiolu (1992), says that a bounded set K is contained in some curve of finite length if and only if a certain “square beta sum”, involving the “width of K” in each element of an infinite system of overlapping “tiles” of descending size, is finite. In this paper we characterize those points of Euclidean space that lie on computable curves of finite length. We do this by formulating and proving a computable extension of the analyst’s traveling salesman theorem. Our extension, the computable analyst’s traveling salesman theorem, says that a point in Euclidean space lies on some computable curve of finite length if and only if it is “permitted” by some computable “Jones constriction”. A...

• Type: Conference
• Year: 2006
• Where: FOCS
• Authors: Xiaoyang Gu, Jack H. Lutz, Elvira Mayordomo
• Keywords: Analyst’s Traveling Salesman, Euclidean Space, FOCS 2006, Theoretical Computer Science, finite Length